Chapter 2 - Fractions I Math Skills – Week 2. Today’s Schedule Turn in Homework Assignment #1 ...

Chapter 2 - Fractions I

Math Skills – Week 2

Today’s Schedule

Turn in Homework Assignment #1 Quiz #1 Lecture on first half of Chapter 2 (Fractions) Stuff:

1. Website Class listing and MyInfo Change your password Post Practice Final exam within the next week

2. Office hour location and time 1. Virtual. Thursdays 6 – 6:45pm

Week 2 - Fractions

Least Common Multiple (LCM) and Greatest Common Factor (GCF) Section 2.1

Introduction to Fractions Section 2.2

Writing Equivalent Fractions Section 2.3

Arithmetic with Fractions (Pt. 1) Addition

Section 2.4 Subtraction

Section 2.5

Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1

The multiples of a number are the products of that number and the whole numbers 1, 2, 3, 4, 5, 6,… Example: The multiples of 2 are:

2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 8 …

Thus the multiples of 2 are 2, 4, 6, 8, …


A number that is a multiple of two or more numbers is called a common multiple of those numbers For Example…8 is a common multiple of 2 and 4.

To find the Lowest Common Multiple (LCM) of a set of numbers use one of the following two methods


Method 1 (Listing multiples) Steps

1. List the multiples of each number2. Identify the common multiples3. Identify which of those is the smallest number.

1. This is the LCM


Example: Find the LCM of 4 and 6 Using Method 1

Step 1: The multiples of 4 are:

4, 8, 12, 16, 20, 24, 28, 32, 36… The multiples of 6 are:

6, 12, 18, 24, 30, 36, 42,… Step 2:

The common multiples of 4 and 6 are: 12, 24, 36,…

Step 3: By inspection, the LCM of 4 and 6 is:

12


Method 2: Using Prime Factorizations Steps:

1. Write the prime factorization of each number

2. Organize these prime factors into a “table of prime factors” (see pg. 65)

3. Circle the greatest product in each column

4. Multiply each of the circled quantities 1. This product is the LCM


Example: Find the LCM of 4 and 6 Use Method 2:

Step 1: The prime factorization of 4 is:

2 x 2 The prime factorization of 6 is:

2 x 3 Step 2: Step 3: Step 4:

LCM = 2 x 2 x 3 = 12

4 =

6 =

2 3

2 x 2

2 3

Organize

Circle greatest products


Which method is better? Which is easier? Tougher example: Find the LCM of 24, 36, and 50

Method 1 Step 1

Multiples of 24 are: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288,

312, 336, 360, 384, 408, 432, 456,…, 1800 Multiples of 36 are:

36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540,…, 1800

Multiples of 50 are: 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550,

600, 650, 700, 750, 800, 850,….., 1800 Step 2/3: LCM 1800


Same example: Find the LCM of 24, 36 and 50 Method 2 Step 1

Prime factorization of 24: 2 x 2 x 2 x 3

Prime factorization of 36: 2 x 2 x 3 x 3

Prime factorization of 50: 2 x 5 x 5

Step 2: Prime Factors Table Step 3: Circle largest products Step 4: LCM is the product of circled quantities

2 x 2 x 2 x 3 x 3 x 5 x 5 = 1800

24 =

36 =

50 =

2 3 5

2 x 2 x 2

2 x 2

2

3

3 x 3

5 x 5


Group Examples: Find the LCM of the following sets of numbers 14, 21

Ans = 42 12, 27, 50

Ans = 2700 Class Examples:

2, 7, 14 Ans = 14

5, 12, 15 Ans = 60

Step 1: Find prime factorization of each numberStep 2: Prime Factors TableStep 3: Circle largest productsStep 4: LCM product of circled quantities

Steps for finding LCM


Recall that the factors of a number are the numbers (1, 2, 3, 4, 5, …) that divide the number evenly

Common factors of a set of numbers are the factors that those numbers have in common.

The Greatest Common Factor of a set of numbers is the largest number in the set of common factors.


To find the GCF of a set of numbers use one of the following two methods Method 1 (Listing factors)

Steps1. List the factors of each number2. Identify the common factors .3. Identify which of those is the largest number. That number is

the GCF Example: Find the GCF of 30 and 105

The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 The factors of 105 are: 1, 3, 5, 7, 15, 21, 35, 105 The Common Factors are 1, 3, 5, 15

The GCF is: 15


Method 2 (Using Prime Factorization) Steps

1. Find the Prime Factorization of each number

2. Write out Prime Factorization table.

3. Circle the smallest product in each column that is not blank

Importante! If column has a blank for one of the numbers, don’t circle anything for that column

4. The product of the circled quantities is the GCF


Example using Method 2: Find the GCF of 90, 168, 420 (Using method 2)

Step 1 The Prime factorization of 90 is:

2 x 3 x 3 x 5 The Prime Factorization of 168 is:

2 x 2 x 2 x 3 x 7 The Prime Factorization of 420 is:

2 x 2 x 3 x 5 x 7 Step 2: Prime Factors table Step 3: Circle Smallest

Product (No Blanks) Step 4: Product of circled numbers is GCF = 2 x 3 = 6

90 =

158 =

420 =

2 3 5

2 x 2 x 2

2 x 2

3

3 x 3

5

7

2

3

7


Group Examples: Find the GCF of the following sets of numbers 12, 18

Ans = 6 21, 27, 33

Ans = 3 Class Examples:

24, 64 Ans = 8

41, 67 Ans = 1

Step 1: Find Prime factorization of each numberStep 2: Prime Factors tableStep 3: Circle Smallest Product in each column (ignore columns with blanks)Step 4: Product of circled numbers is GCF

Steps to find GCF

Introduction to Fractions – Section 2.2

A fraction is the representation of a specified portion of a whole number.

24

14

34

44

Numerator

Denominator

Fraction Bar


Definitions Proper Fraction is a fraction that is less

than 1 Numerator is smaller than the denominator

Mixed number is a number greater than 1 Whole number part and a fractional part.

Improper Fraction is a fraction greater than or equal to 1

Numerator is greater than the denominator

34

341

74


Convert Improper fractions Mixed numbers Steps

1. Divide the Numerator into the Denominator2. Fractional Part: Write any remainder as a fraction by placing it

over the original denominator Example: Write 13/5 as a mixed number

Convert Mixed numbers Improper Fractions Steps

1. Multiply the denominator of the fractional part by the whole number part

2. Add this product to the numerator3. Write the sum from step 2 over the denominator of the fractional

part Example: Write 7 3/8 as an improper fraction


Class Examples: Write 22/5 as a mixed number

4 2/5 Write 28/7 as a whole number

4 Write 14 5/8 as an improper fraction

117/8 Write 10/3 as a mixed number

3 1/3

Writing Equivalent Fractions – Section 2.3

Equivalent fractions are equal fractions that look different

Example 4/6 is equivalent to 2/3

Remember the ones property in multiplication? 1 x Number = Number

Agree? 2/3 x 1 = 2/3 2/3 x 1/1 = 2/3 2/3 x 4/4 = 2/3 = 8/12 2/3 x 5000/5000 = 2/3 = 10000/15000


Example: (Finding equivalent fractions) What is an equivalent fraction to 5/8 that has a denominator of 32?

Ask yourself…self…what do I have to multiply the denominator of 5/8 by to get 32?

Or you could just divide 32 by 8 4

5/8 x 1 = 5/8 x 4/4 = 20/32 20/32 is a fraction with 32 in the denominator that is equivalent

to 5/8 Another example

Write 2/3 as an equivalent fraction that has a denominator of 42 Divide 42 by 3 = 14

2/3 x 14/14 = 28/42 is equivalent to 2/3 Example write 4 as a fraction with 12 in denominator


Class Examples: Write 3/5 as an equivalent fraction with a

denominator of 45 Fill in the blank

1. ½ = __ /32

2. 2/3 = __ / 12

3. 6 = __ / 11


A fraction is in simplest form when the numerator and denominator have no common factors (other than 1) Example: 4/6 written in simplest form is 2/3

To write a fraction in simplest form Steps

1. Write prime factorization of the numerator and denominator

2. Cancel (divide) out all common factors. 1. Remaining products are the new Numerator and

Denominator


Examples Write 15/40 in simplest form

= 3 x 5 / 2 x 2 x 2 x 5 = 3/8 Write 6/42 in simplest form

= 2 x 3 / 2 x 3 x 7 = 1/7 Write 30/12 in simplest form

2 x 3 x 5 / 2 x 2 x 3 = 5/2 = 2 1/2


Class Examples: Write the following in simplest form

16/24 = 2 x 2 x 2 x 2 / 2 x 2 x 2 x 3 = 2/3

8/56 2 x 2 x 2 / 2 x 2 x 2 x 7 = 1/7

15/32 = 3 x 5 / 2 x 2 x 2 x 2 x 2 = 15/32

48/36 = 2 x 2 x 2 x 2 x 3 / 2 x 2 x 3 x 3 = 4/3 = 1 1/3

Addition of Fractions and Mixed Numbers 2.4

The key is the denominator. To add fractions together, each fraction must have the same denominator.

If the denominators are the same Steps

1. Add the Numerators

2. Place the sum of the

Numerators over the

common denominator

1. Write the sum in simplest form

512

+1112

1612

= 43

=


If denominators are not the same: Steps

1. Find the Lowest Common Denominator (LCD) of the two fractions Note: this quantity is the LCM of the denominators

2. Rewrite each fraction as an equivalent fraction with the LCD as the denominator.

3. Add the numerators

4. Place this sum over the common denominator Example: 1/2 + 1/3 = ?

LCM = 6, then 3/6 + 2/6 = 5/6


More Examples: Find 7/12 more than 3/8

Lowest Common Denominator (LCD) = 24 14/24 + 9/24 = 23/24

Add 5/8 + 7/9 LCD = 72

45/72 + 56/72 = 101/72 = 1 29/72

Add 2/3 + 3/5 + 5/6 LCD = 30

20/30 + 18/30 + 25/30 = 63/30 = 2 3/30 = 2 1/10


Class Examples: Find the sum of 5/12 and 9/16

LCM = 48 20/48 + 27/48 = 47/48

Add 7/8 + 11/15 LCM = 120

105/120 + 88/120 = 193/120 = 1 73/120

Add 3/4 + 4/5 + 5/8 LCM = 40

30/40 + 32/40 + 25/40 = 87/40 = 2 11/40


Addition of mixed numbers Steps

1. Find the Lowest Common Denominator (LCD) of the two fractions Note: this quantity is exactly the (LCM) of the

denominators

2. Add the fractional parts3. Add the whole number parts4. Put fractional part in simplest form

Example: what is 6 14/15 added to 5 4/9 ? LCM = 45, then5 20/45 + 6 42/45 = 11 62/45

= 11 + 1 17/45 = 12 17/45


More Examples: Find 5 more than 3/8

LCD = Don’t need this 5 3/8

Add 17 + 3 3/8 LCD = Don’t need this

20 3/8

Add 5 2/3 + 11 5/6 + 12 7/9 LCD = 18

5 12/18 + 11 15/18 + 12 14/18 = 28 41/18 = 30 5/18


Class Examples: Find the sum of 29 and 7 5/12

LCD = Don’t need this 46 5/12

Add 7 4/5 + 6 7/10 + 13 11/15 LCD = 30

7 24/30 + 6 21/30 + 13 22/30 = 26 67/30 = 28 7/30 Add 9 3/8 + 17 7/12 + 10 14/15

LCD = 120 9 45/120 + 17 70/120 + 10 112/120 = 36 227/120

= 37 107/120


Word problems discussion Pg 80, You Try It 9

Add all time spent together Pg 80, You Try It 10

Add all time spent working overtime Multiply total time spent working overtime by the

overtime hourly rate.

Subtraction of Fractions and Mixed Numbers 2.5

Again…the key is the denominator. To subtract fractions, each fraction must have the same denominator.

If the denominators are the same Steps

1. Subtract the Numerators

2. Place the difference of

the new numerators

over the common denominator

1. Write the difference in simplest form

1112

-5

127

12=


If denominators are not the same: Steps

1. Find the Lowest Common Denominator (LCD) of the two fractions Note: this quantity is exactly the Least Common Multiple

(LCM) of the denominators

2. Rewrite each fraction as an equivalent fraction with the LCD as the denominator.

3. Subtract the numerators

4. Place this difference over the common denominator Example: 5/6 – 1/4 = ?

LCM = 12, thus 10/12 - 3/12 = 7/12


More Examples: Subtract 3/4 - 2/5

LCD = 20 15/20 – 8/20 = 7/20

Subtract 53/60 - 7/12 LCD = 60

53/60 – 35/60 = 18/60 = 3/10

11/16 – 5/12 = ? LCD = 48

33/48 - 20/48 = 13/48


Class Examples: Subtract 5/6 – 4/15

LCD = 30 25/30 – 8/30 = 17/30

Subtract 13/18 – 7/24 LCD = 72

52/72 – 21/72 = 31/72


Subtraction of mixed numbers Steps

1. Find the Lowest Common Denominator (LCD) of the two fractions Note: this quantity is the LCM of the denominators

2. Subtract the fractional parts1. Borrow if necessary

Borrow 1 from the whole number part and rewrite it as an equivalent fraction to 1 using with the same LCD

3. Subtract the whole numbers


Subtraction of mixed numbers Example: (No Borrowing) what is 5 5/6 subtracted from 2 3/4?

LCD = 12 5 10/12 – 2 9/12 = 3 1/12

Example: (With Borrowing): Subtract 5 – 2 5/8 LCD = Don’t need it

4 8/8 – 2 5/8 = 2 3/8 Example: (With Borrowing): Subtract 7 1/6 – 2 5/8

LCD = 24 7 4/24 - 2 15/24 = 6 28/24 – 2 15/24 = 4 13/24


More Examples: Subtract 15 7/8 – 12 2/3

LCD = 24 15 21/24 – 12 16/24 = 3 5/24

Subtract 9 – 4 3/11 LCD = Don’t need this

8 11/11 – 4 3/11 = 4 8/11

Find 11 5/12 decreased by 2 11/16 LCD = 48

11 20/48 – 2 33/48 = 10 68/48 – 2 33/48 = 8 35/48


Class Examples: Subtract 17 5/9 – 11 5/12

LCD = 36 17 20/36 – 11 15/36 = 6 5/36

Subtract 8 – 2 4/13 LCD = Don’t need this

7 13/13 – 2 4/13 = 5 9/13

Find 21 7/9 minus 7 11/12 LCD = 36

21 28/36 – 7 33/36 = 21 64/36 – 7 33/36 = 14 31/36


Word problems discussion Pg 88,

6 Add all time spent together

You Try It 7 How would you approach this problem?

Add all the weight lost over the first two months 13 ¼ pounds lost in the first two months Subtract 13 ¼ from the total of 24. (10 ¾ pounds

left)

Chapter 2 - Fractions I Math Skills – Week 2. Today’s Schedule Turn in Homework Assignment #1 ...

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